Many techniques are known in many technical fields for controlling the response or behaviour of a device or system. In Control Theory literature, the term “plant” is generally used to refer to such a device whose behaviour we are trying to control. In this context, a plant could be anything from the tiny armature that carries the cutting stylus of an LP disc-cutting head to a space-rocket whose motion we are sensing and trying to control by sending instructions to its motors. Other examples include the power output stage of an amplifier and a mechanical actuator. Indeed, the concept can be extended to include control of a complete factory or chemical processing plant.
A substantial part of the control theory literature addresses the control of systems, for example biological systems, in which the effect of adjusting a control input may be extremely uncertain a priori, so that the controller must estimate the linear and nonlinear transfer functions of the plant starting from very little a priori information. The present invention is directed more towards systems whose behaviour is known well enough for a reasonable model to be constructed.
Predistortion is a technique whereby an imperfect ‘plant’ P is preceded by a corrector that approximates its inverse P−1, as shown in FIG. 1. Similarly, there is also postdistortion in which the corrector is placed afterwards. Either method begs the question of how the inverse P−1 can be calculated or implemented. Gerzon made a fundamental contribution in 1972 by exhibiting a method whereby an approximate inverse could be constructed provided that multiple instances or replicas of P could be made available (Gerzon, M. A., “Predistortion Techniques for Complex but Predictable Transmission Systems”, J. Audio Eng. Soc., Volume 20, pp. 475-482, July 1972). For example, in the case of the audio distortion produced by bandlimiting in the Intermediate Frequency (IF) section of an FM radio receiver, Gerzon showed how the inverse distortion could be produced using one or more low-power copies of the actual FM transmission system. Gerzon's approximate inverse would then be placed prior to the FM modulator in the actual radio transmitter. Today, Gerzon's approximate inverse would typically be implemented in digital signal processing (DSP) software, the “low-power copies” being implemented as software modules that model the transmitter and receiver combination.
Feedback is a technique whereby the input to the plant is modified in dependence on its output. An example is shown in FIG. 2, where a system input feeds the plant P, which provides the system output. A subtraction node takes the difference between the system's output and its input, thereby deriving an error signal that is filtered by feedback filter G and then fed to the plant's input in a polarity such as to reduce the error. In the example shown, a subtraction node is employed. If the plant has several inputs and output, G may be a matrix filter, but for simplicity we shall concentrate on the 1-channel case.
There are many feedback topologies in the electronics literature, some of which have been described as ‘novel’. In fact, as long as it is the plant P that provides the system output and P is the only nonlinear component in the system, all these topologies are equivalent modulo overall frequency response. That is to say, any one topology can be made equivalent to any other by suitable adjustment of the transfer functions of the various blocks, and maybe by adding a linear filtering stage at the input to provide the desired system transfer function.
The potential benefits of feedback include: ironing out irregularities in the plant's frequency response and reducing nonlinearities and noise introduced by the plant. The classic problem of feedback design is to apply enough feedback for the desired performance improvement while keeping the system stable. The ‘control theory’ literature considers this problem at length. In the linear case, the problem is amenable to mathematical analysis, though the equations may nevertheless be daunting.
Most introductory books on control theory devote one or at most two chapters to the nonlinear case. In dealing with weakly nonlinear systems, engineers frequently start from an assumption of linearity, but allow some margin (“phase margin” or “gain margin”) in the hope that deviations from the linearity will not be large enough to send the system unstable. Other factors that may be allowed for in assessing the stability margin are unknown external conditions (e.g. load), component tolerances and component drift with temperature and/or time.
The techniques of predistortion and feedback both have associated strengths and weaknesses. Predistortion can correct for known nonlinearity of the plant and does not have stability problems, but is powerless to deal with deviations from the expected plant behaviour. Feedback can correct both known and unknown nonlinearity of the plant, and can correct other deviations from the expected behaviour, but may be sent unstable if the nonlinearities or other deviations are too large.
A natural extension to these techniques is to use a combination of predistortion and feedback. FIG. 3 illustrates an example of this in which the principle being applied may be summarised as “correct what you can by means other than feedback, then apply overall feedback to tidy up what remains”. Here, P−1 has a direct effect in reducing the distortion that appears on the output, and also potentially an indirect effect because the improved linearity of the combination (P−1.P) may make it possible to apply more feedback before instability sets in.
This technique can be extremely effective when it works, but in some cases P may not have a causal inverse. Provided P's nonlinearity is not too strong, the Gerzon procedure may still be used but now it returns a pseudo-inverse P−1 such that (P−1.P) approximates a pure delay. Indeed, successively more accurate higher-order approximations generated by the Gerzon procedure incur more and more delay. In a feedback system, loop delay is usually the prime determining factor that limits the amount of feedback that can be stably applied. Thus, with the delay introduced by the Gerzon predistortion, the system shown in FIG. 3 may be unable to support sufficient feedback to correct adequately the unknown disturbances that are not corrected by P−1. To restate the problem, the plant P may itself have a delay, and there is nothing that can be done about that. It will limit the feedback. The unfortunate consequence of the system shown in FIG. 3 is that P−1 may further increase the delay.
FIG. 4 shows another prior-art feedback topology that may be employed. The operation of the system shown in FIG. 4 may be understood by considering that any error e committed by P is isolated by the subtraction node at the bottom of the figure and, after filtering with filter H, is subtracted from the input by means of an addition unit with negative polarity input. Assuming that H has unity gain within the frequency range of interest, and that P also approximates unity gain, the error is approximately cancelled at the output.
This “inner form” topology is widely used in noise shaping, in which the “plant” P is actually a quantiser, and the reason for the feedback is to reduce quantisation noise (or at least, to reduce it within a particular frequency range). This is a pure digital application, in which linearity is taken for granted and stability can be ensured (subject to start-up conditions and adequate arithmetic headroom) by a purely linear analysis.
In a simplest “first order” noise shaper, H is just a z−1 unit delay, which is needed to avoid a delay-free feedback loop. Such a unit delay has unity gain at DC, but has a phase rotation that increases as frequency increases. Much effort has been applied to designing digital filters H that incorporate the unit delay but then compensate the phase rotation so as to provide a close approximation to unity gain over some desired frequency range. However, it should be noted that, in order provide effective cancellation of error over a desired frequency range, it is sometimes necessary to give H a large gain at frequencies outside that frequency range.
In view of the prior art techniques described above, there is a need for a robust technique that combines advantages of predistortion and feedback and is able to correct for both known and unknown nonlinearities, particularly where the nonlinearity is large enough to cause a conventional control system to go unstable.